We report results from extensive new XMM- Newton observations of the ultraluminous X-ray source (ULX) NGC 5408 X-1, one of the few ULXs to show quasi-periodic X-ray variability. We detect quasi-periodic oscillations (QPOs) in each of four new (approximately equal 100 ks each) pointings, expanding the range of frequencies and rms amplitudes observed from the source to 10-40 mHz and 10-45 %, respectively. However, similarly significant variations in the power-law photon spectral index, Gamma, are not observed. We use the results of timing and energy spectral modeling to compare with the timing and spectral correlations seen in stellar-mass systems. We find that the qualitative nature of the timing and energy spectra of NGC 5408 X-1 are very similar to stellar-mass black holes in the steep power-law state exhibiting Type-C QPOs. However, in order for this analogy to quantitatively hold we must only be seeing the so-called saturated portion of the QPO frequency - photon index (or disk flux) relation. Assuming this to be the case, we place a lower limit on the mass of NGC 5408 X-1 of approx greater than 800 Solar Mass. Alternatively, the QPO centroid frequency is largely independent of the spectral parameters, in which case a close analogy of NGC 5408 X-1's mHz QPOs with Type-C QPOs in stellar systems is problematic. Measurement of the source's timing properties over a greater range of spectral parameters (in particular the spectral index) is needed in order to definitively resolve this ambiguity. We searched all the available data for both a broad Fe emission line as well as high frequency QPO analogs (0.1 - 1 Hz), but detected neither. We place upper limits on the equivalent width of any Fe emission feature in the 6 - 7 keY band, and of the amplitude (rms) of a high frequency QPO analog of approx equal 10 eV and approx equal 4%, respectively.